LA 333 



■If ' : ''W %WM 



MONOGRAPH 



No. 3 



Arithmetic Survey 

Newark, New Jersey 



Harvard University, 

Library of the Graduate School 

of Education 



Arithmetic Survey 

in the 

Public Schools 

of 
Newark, N. J. 




BOARD OF EDUCATION 






NEWARK, NEW JERSEY 
DECEMBER 1919 



i A 333 

-fa As 
i?if 



LIBRARY OF CONGRESS 



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Table of Contents 

PAGE 

Foreword 5 

Introduction 7 

Chart I — Accuracy Tests, Newark Compared with Woody Standards 8 

Woody Tests, Series B 9 

Chart II — Newark, Four Fundamentals Combined, Compared with 

Woody Standards Combined 10 

Stone Reasoning Test 11 

Chart III — Reasoning Tests, Newark Compared with Stone Test 

and with Three Cities 12 

Table I — Scores Made by Each School in Newark in the Woody 

Tests, Series B, and in the Stone Reasoning Test. . .opp. p. 13 

Analysis of Conditions as Shown by Table 1 13 

Table II — Comparison of Groups of Schools 15 

Distribution of Pupils' Scores 17 

Chart IV — Distribution of Scores Attained in Addition, Multiplica- 
tion, and Reasoning °PP- P- 17 

Chart V — Per Cent, of Accuracy in Each of Four Fundamentals. . . 18 

Accuracy 19 

Table III — Per Cent, of Accuracy in the Four Fundamentals 19 

Discussion of Individual Problems 19 

Addition Scale 20 

Table IV — Addition Problems Incorrect or Not Attempted 21 

Subtraction Scale 22 

Table V — Subtraction Problems Incorrect or Not Attempted 23 

Multiplication Scale 24 

Table VI — Multiplication Problems Incorrect or Not Attempted.. . 25 

Division Scale 26 

Table VII — Division Problems Incorrect or Not Attempted 27 

Reasoning Test 28 

Conclusions 30 



Foreword 

The survey in penmanship in the schools of Newark proved to be 
so suggestive and valuable that, upon the recommendation of the Super- 
intendent of Schools, a survey of arithmetic was authorized by the Com- 
mittee on Instruction and Educational Supplies. 

This survey was made under the direction of Mr. Elmer K. Sexton, 
Assistant Superintendent in charge of the Department of Reference and 
Research. Mr. Sexton's report of the survey, which follows, is very 
suggestive and shows clearly wherein classroom work in the subject is 
weak or strong and where the emphasis should be placed to improve 
weak work. It also indicates needed adjustments in the course of study. 

Among the important facts revealed by the survey is the satisfactory 
condition in grades one to four inclusive, where excellent results are 
obtained in the fundamentals. It is fair to infer that, in the main, the 
course of study for these grades is well adapted to the ability of the 
children, and that the methods of teaching are effective. The record of 
the fifth grade is also gratifying. The great contrast between the results 
in the fifth and the sixth grades is striking. This is probably due to the 
fact that the work of the sixth grade is less interesting, there is little 
that is strong in its appeal, and the operations in denominate numbers 
constitute the new features of the work. 

The survey reveals the need of continued drill in the fundamentals 
in the upper grades. The weakness shown in the formal work is due to 
the fact that in these grades the problem work, or so-called "thought- 
work," requires a very large part of the time. Neither phase of the 
work can safely be neglected, because the children are at that period of 
life when habits are in the forming. The habit of doing accurate work 
in the fundamentals is of such great value that it must be fixed. There 
must be a better balance between the formal and the thought work in 
the upper grades. 

DAVID B. CORSON, 

Superintendent of Schools. 



Report of Arithmetic Survey 

On April 21, 1919, the Superintendent recommended to the Com- 
mitee on Instruction and Educational Supplies that a survey be made 
of the Newark schools in the subject of arithmetic. The Board approved 
the recommendation, and included in the resolution authority for making 
a survey in spelling. 

Tests in arithmetic were given in all schools of the city May 29 to 
June 2, 1919, inclusive. The tests given were the Woody Scales, Series 
B, and the Stone Reasoning Test. In order that they should be given as 
uniformly as possible and to the greatest advantage to the pupils, 
teachers were selected to conduct the tests who were well qualified to 
conduct them in an efficient manner and to secure from the pupils their 
best work. A teacher was sent to each school (in no case the school 
where she regularly taught), where she conducted all of the tests for 
that school. Uniform directions were given to these teachers covering 
the procedure for giving the tests. The test papers were then sent to 
the Research Department, where they were scored and the results tab- 
ulated. For the purpose of comparing the work done in the various 
schools, all pupils in the 4A, 6A, and 8A -grades were tested, except in 
those schools having four classes of one grade, in which case the pupils 
in only two of these classes were tested. A sufficient number of pupils 
of the other grades between 4A and 8A were tested to secure an accurate 
median in these grades to complete the line of the graphs for all grades 
above the 4B, in order that the work of this city might be compared with 
the work of other cities. From the 60,000 papers obtained as a result of 
these tests a sufficient number (31,405) were scored to give an accurate 
measure of the schools of the city for purposes of comparison. 

The principal aims in mind in making the tests were : 

1 . To compare the work of the Newark schools in arithmetic 
with the work in other cities. 

2. To compare the work in the various schools of Newark. 

3. To observe, if possible, the efficiency of groups of schools 
having peculiar types of children or peculiar organization. 

4. To ascertain as far as possible where the weaknesses lie, both 
as to school, grade, or phase of subject, and as to methods of 
instruction. 

The professional spirit in which this survey has been received is 
highly commendable and indicates that all teachers and principals are 
very willing to have the work of their pupils measured and to learn 
wherein their school or class is weak or strong. It means that improve- 
ment must inevitably follow. 

Conclusions concerning a certain school should not be drawn too 
quickly, as this is a survey in one subject only, but in a subject which 



19 



PUBLIC SCHOOLS OF NEWARK, N. J. 

Chart I 

Woody Accuracy Tests * A/ewarA jr June, ,9/9 
Compared tv/ffy Woody Standards 



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Newark resu/fs shown by ba/f year grades- 
Woody resu/fs shown Ay -fo // year ■ y redes 



H-B V-A SB SA 65 <*A IB ja 8B &a 



ARITHMETIC SURVEY 



lends itself easily to measurement. The facts in the case should be 
followed with an open mind in order that the school system may reap 
the maximum benefit from this survey. 

The results are partly exhibited by means of graphs and tables. 



WOODY TESTS, SERIES B 

Chart I represents the ability of Newark pupils in abstract work 
in the four fundamental operations in arithmetic, together with 
the Woody standards as ascertained by a very careful examination of 
nearly 5,000 pupils conducted in the main by Mr. Woody himself. It 
may be said, however, that in giving this test and securing these stan- 
dards, Mr. Woody placed no time limit upon the pupils while they were 
working, whereas ten minutes was given to the Newark pupils. This, 
of course, would cause a lowering of the Newark standards, as compared 
with the Woody standards. The question might arise whether or not 
ten minutes is not ample time in which to do this work, if the pupils 
have been well trained in the four fundamental operations, except, per- 
haps, in the lower grades where pupils are given some examples covering 
subjects with which they are not yet familiar. Good work in the funda- 
mentals is almost always done rapidly. The Woody tests were made 
in the early part of the year, the Newark tests in the latter part, so the 
Woody standard is shown on a line with the B grades rather than the A. 

The chart shows that addition in the Newark schools is the poorest 
of the four fundamental operations when compared with the Woody 
standard and that multiplication is the best. Addition is a very difficult 
process to teach unless the work is organized, and systematically devel- 
oped and drilled upon. Much more drill is of necessity given the multi- 
plication combinations in all subsequent mathematical work than is 
given the addition combinations. It is easier to put the multiplication 
tables into; practice through figures than the addition combinations, and 
consequently pupils can read through a multiplication example much 
more rapidly than down a column of figures. 

The line in addition begins slightly above the Woody standard in 
the 4B, crosses it between the 4B and 4A, and remains below with an 
ever-widening gap through the upper grades. The line of subtraction 
has a varying course and finally falls below the Woody standard in the 
7A and continues below through the eighth grade. The multiplication 
line crosses the Woody standard between the 5A and 6B, and remains 
below until it reaches the 8A, where it surpasses the Woody standard. 
The result in division is shown to surpass that found by the Woody 
test in the lower grades tested to a greater degree than any of the other 
operations. The line crosses the Woody standard between the 6B and 
6A grades and finishes somewhat below. It will be observed that, in 
the lower grades, the work in the Newark schools surpasses the Woody 
standards in all of the four fundamental operations. This would tend 
to show that the primary grades of the city have been doing their work 
well, exceptionally well, while the upper grades have not kept up the 
good standing. 



10 



PUBLIC SCHOOLS OF NEWARK, N. J. 



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ARITHMETIC SURVEY 11 

The information found on Chart I has been combined for Chart II, 
which shows the work of the Newark schools in the four fundamental 
operations combined, as compared with the combined Woody standards 
in the four fundamental operations, together with the percentage of 
accuracy of Newark pupils in fundamentals. This brings out more 
graphically the facts which a careful study of Chart I shows, viz. : that 
the work of the primary grades in the Newark schools is well done in 
the four fundamentals and that the work continues better than the 
Woody standard until between the 5A and 6B, where the line falls 
below the Woody standard and continues below through the eighth 
grade. (See also Chart III, where the reasoning line falls below in 
nearly the same place.) It is also quite evident from both charts that 
all "A" classes are stronger than the "B" classes. There seems to be no 
explanation for this unless it is psychological. It may take less mental 
effort for a teacher to promote from 5B to 5A than from 5A to 6B, where 
there is a change of grade. 

The 5A grade appears to be the strongest grade of all those tested. 
This is rather surprising, because the 5B and 5A teachers have long 
thought that the work laid down for them in the course of study was 
rather heavy. The line drops slightly between the 4A and 5A, but the 
line to the 5B is very nearly straight with the 6B and subsequent grades. 
The decided drop may be partly due to the very high standing of the 
5A in all tests. It is very noticeable that the 5A grade in every opera- 
tion stands higher in proportion than any other grade examined. This, 
of course, makes the contrast with the 6B and subsequent grades very 
much greater than it otherwise would be. 

The percentage of accuracy, as shown on Chart II, follows very 
closely the results in the fundamentals in the various grades. The ques- 
tion may arise whether accuracy leads to good results or good results 
lead to accuracy. At any rate, it is evident that they are closely corre- 
lated. It may be observed, also, that the 5A grade represents the highest 
relative point in accuracy of all the grades tested, while the falling off of 
accuracy in the 6B, 6A, and 7B is not so marked as in the results in the 
four fundamentals. 



STONE REASONING TEST 

Chart III presents the results of the reasoning test compared with 
the average of three cities where the test has been given by outside and 
disinterested experts — Salt Lake City, San Francisco, and Butte— and 
with Salt Lake City and San Francisco the two cities nearest the size 
of Newark. Salt Lake City presents a better showing than Newark and 
is highly conmmended in the report of the committee, of which Mr. 
Elwood B. Cubberley, of Leland Stanford, Jr., University, was chair- 
man. He says, "From these results it is clear that the schools of this 
city rank high in the ability of their children to reason." 

The reasoning ability of the Newark pupils is well up to the Salt 
Lake City standard and somewhat above the average of the three cities 
up to the 6B grade. It then falls rapidly in 6A and 7B below the three- 



12 



PUBLIC SCHOOLS OF NEWARK, N. J. 



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ARITHMETIC SURVEY 13 

city average, but returns in 7 A, 8B and 8A, finishing much above the 
three-city average, but still below the Salt Lake City standard, and 
makes a greater per cent, of increase than any of the three cities. 

The Newark schools have a high record in the reasoning test 
when compared with the three cities, although it is not as good as that 
of Salt Lake City. The per cent, of gain in reasoning power from the 
5B to 8A is as follows: Salt Lake City 184%, Newark 153%, San 
Francisco 143%, average of the three cities, 186%. The lower per cent, 
advancement in Newark is largely due to the excellent start in the lower 
grades and the subsequent loss in the 6A and 7B grades. 

By comparing Chart II and Chart III we find that the difficulty, 
both in the fundamental operations and in the reasoning, occurs in 
nearly the same grades, that the pupils come up from the primary school 
through the fifth grade in excellent condition, but that during the next 
few half-year grades there is a decided falling off, strengthening again 
as the line approaches the eighth grade. The decided bend in the lines 
of the two graphs should be carefully considered. The cause of this 
loss of efficiency may be found in the course of study or in the topics 
treated in these grades which perhaps have not been so thoroughly 
organized as to subject matter, nor as systematically drilled upon as 
those of other grades. When this deficiency is removed Newark will 
stand very high in both formal work and in reasoning. 

ANALYSIS OF CONDITIONS AS SHOWN BY TABLE I 

Table I gives the results in the 4A, 6A, and 8A grades of the four 
fundamental operations and the reasoning test in the 6A and 8A grades. 
The figures representing these results have also been totaled for pur- 
poses of general comparison. The Woody and the Stone standards are 
given at the top of the table, together with the Newark standard. From 
this table each principal can compare the work of his school with that 
of the other schools in the four processes for each grade given, and in 
reasoning for the 6A and 8A grades. He can find which grade is weakest 
or which process is weakest. If a grade is weak, he can learn which 
process of that grade is weak ; or, if a process is weak, he can learn in 
which grade the weakness lies. In short, he can locate strong or weak 
points and give less or more time to them, improve the methods of 
instruction, or more carefully organize his drill lessons, as seems neces- 
sary. 

It will be seen that school number 19 is below the Newark median 
in all four operations in the 4A grade, while it is well above in the 6A 
and 8A grades. The 4A grade in this school is taught by an inexpe- 
rienced teacher who has not yet been able to control the class, and the 
pupils have not only not improved in their work, but have actually 
learned very bad habits. This emphasizes the importance of good 
teachers. 

Some schools show a decided improvement in the four fundamentals 
from the 4A to the 8A grade, as found in schools number 42 and 36. 
School number 42 ranks 19th in 4A, 16th in 6 A, and 1st in 8A. School 



TABLE I 

Table of Scores Made by Each School in Newark in the Woody Tests 

Series B and in the Stone Reasoning Test 

Tune. 1919 



Woody Median. 


11.0 


8.0 


7.0 


5.0 




16.0 


12.0 


15.0 


10.0 




18.5 


14.5 


18.0 


14.0 






STONE REASONS 


Q TEST 












12.06 


9.46 


11.16 


7.54 


40.18 


15.13 


12.63 


15.49 


11.37 


54.45 


16.73 


14.36 


18.19 


13.33 


62.31 




3.27 


6. SO 


Average 






Oitlei 




4 A 






6 A 










8 


A 






5.27 


8.60 


Newark 
Aver- 
age 


































Grand 










A 


S 


M 


D 


Total 


A 


S 


M 


D 


Total 


A 


S 


M 


D 


Total 


Total 


6A 


8A 


Total 


School No. 1 


12.67 


1.80 


11.56 


8.39 


34.42 


12.33 


11.79 


13.56 


10.50 


4S.1S 


15.50 


14.56 


17.33 


13.72 


61.11 


143.71 


4.84 


9.29 


14.13 


2 


11.20 


9.12 


9.17 


5.00 


34.49 


14.80 


10.92 


14.71 


9.38 


49.S1 


15.35 


13.18 


16.*0 


11.79 


56.82 


141.12 


4.22 


6.81 


11.03 


3 


11.92 


9.47 


11.50 


7.89 


40.78 


12.22 


10.11 


12.50 


8.28 


43.11 


16.14 


12.50 


17.20 


11.23 


57.07 


140.96 


2.31 


7.15 


9.46 


4 


11.77 


9.83 


10.75 


8.43 


40.78 


14.33 


11.96 


15.00 


10.86 


52.15 


15.25 


13.70 


16.93 


12.55 


58.43 


151.36 


6.45 


S.85 


15.30 


5 


12.13 


9.69 


10.63 


5.65 


38.10 


15.65 


12.42 


14.75 


12.25 


55.07 


16.40 


13.92 


1S.38 


13.07 


61.77 


154.94 


4.90 


6.36 


11.26 


6 


11.50 


9.47 


11.17 


8.59 


40.73 


14.44 


11.71 


14.00 


10.83 


50.98 


16.00 


14.15 


15.90 


12.88 


58.93 


150.64 


4.74 


8.35 


13.09 


8 


12.50 


9.58 


10.78 


8.18 


41.04 


14.50 


11.71 


15.44 


11.08 


52.73 


16.S6 


14.45 


17.17 


12.69 


61.17 


154.94 


4.85 


8.79 


13.64 


9 


12.97 


10.80 


12.12 


9.88 


45.77 


15.23 


13.17 


15.40 


12.40 


56.20 


17.43 


14.55 


18.00 


13.S8 


63.86 


165.83 


4.80 


10.28 


15.08 


10 


11.38 


9.45 


11.36 


7.17 


39.36 


15.77 


11.38 


16.80 


12.36 


56.31 


16.69 


15.17 


18.17 


14.06 


64.09 


159.76 


6.18 


11.96 


18.14 


11 


12.11 


9.11 


11.22 


7.79 


40.23 


15.97 


13.18 


16.33 


12.00 


57.48 


16.59 


14.83 


18.71 


13.29 


63.42 


161.13 


5.86 


11.18 


17.04 


12 


13.31 


9.74 


12.09 


8.33 


43.47 


16.00 


13.00 


16.35 


11.88 


57.23 


17.10 


14.69 


18.17 


14.05 


64.01 


164.71 


6.41 


10.07 


17.38 


13 


11.65 


9.61 


10.88 


8.06 


40.20 


14.70 


13.21 


15.23 


11.56 


54.70 


17.08 


14.73 


17.96 


13.55 


63.32 


158.22 


5.82 


10.59 


16.41 


15 


11.81 


9.64 


11.19 


8.52 


41.16 


15.3S 


12.25 


15.56 


11.50 


54.69 


17.25 


14.41 


18.45 


14.16 


64.27 


160.12 


4.60 


10.31 


14.91 


16 


12.08 


9.81 


11.30 


7.50 


40.69 


16.50 


12.42 


16.15 


11.50 


56.57 


18.13 


14.20 


17.50 


12.67 


62.50 


159.76 


5.14 


8.49 


13.63 


17 


11.9a 


9.48 


11.32 


8.63 


41.36 


15.50 


13.11 


15.93 


11.50 


56.04 


17.14 


14.81 


18.21 


13.36 


63.52 


160.92 


5.78 


9.96 


15.74 


18 


12.46 


9.96 


11.42 


8.50 


42.34 


16.07 


13.32 


17.14 


13.00 


59.53 


17.60 


15.00 


19.29 


14.11 


66.00 


167.87 


6.90 


12.78 


19.68 


19 


11.10 


8.75 


10.S3 


5.59 


36.27 


16.50 


14.22 


15.83 


12.17 


58.72 


17.18 


14.47 


1S.80 


13.44 


63.89 


158.88 


4.86 


9.59 


14.45 


20 


10.33 


9.25 


10.75 


7.75 


38.0S 


14.83 


11.69 


13.00 


10.25 


49.77 


16.67 


14.07 


17.17 


13.25 


61.16 


149.01 


4.02 


9.20 


13.22 


22 


12.10 


9.65 


11.93 


9.57 


43.25 


15.63 


12.95 


15.30 


11.60 


55.48 


16.43 


14.61 


18.68 


13.89 


63.61 


162.34 


5.52 


11.56 


17.08 


23 


12.53 


9.53 


11.08 


8.50 


41.64 


16.50 


12.92 


16.30 


11.88 


57.60 


16.89 


14.85 


19.15 


13.75 


64.64 


163.88 


7.21 


11.41 


18.62 


24 


12.07 


9.54 


10.75 


7.50 


39.86 


15.27 


13.42 


15.41 


11.30 


55.40 


15.64 


13.44 


18.07 


12.75 


59.90 


155.16 


4.71 


8.37 


13.08 


25 


12.15 


9.40 


11.94 


6.44 


39.93 


15.47 


13.00 


16.00 


11.67 


56.14 


16.75 


14.75 


18.67 


13.71 


63.88 


159.95 


5.78 


8.64 


14.42 


27 


10.96 


9.48 


10.89 


7.00 


38.33 


15.29 


13.40 


16.63 


11.50 


56.82 


17.06 


14.13 


18.21 


12.81 


62.21 


157.36 


5.11 


8.01 


13.42 


28 


11.57 


9.63 


11.43 


6.41 


39.04 


15.33 


13.50 


15.83 


11.55 


56.21 


17.50 


14.75 


18.39 


13.72 


64.36 


159.61 


5.86 


10.95 


16.81 


30 


10.53 


9.52 


10.80 


6.93 


37.78 


14.29 


12.06 


14.92 


10.46 


51.73 


16.25 


14.29 


18.17 


13.13 


61.84 


151.35 


4.09 


9.02 


13.11 


31 


12.67 


8.67 


10.73 


7.00 


39.07 


14.00 


11.60 


15.90 


11.33 


52.83 


16.3S 


13.75 


18.30 


13.25 


61.68 


153.58 


5.27 


9.90 


15.17 


33 


13.03 


9.69 


11.68 


8.00 


42.40 


16.15 


12.94 


17.19 


11.94 


58.22 


17.55 


14.70 


18.60 


12.91 


63.76 


164.38 


5.79 


10.56 


16.35 


85 


13.22 


9.63 


10.85 


8.43 


42.13 


15.64 


13.85 


15.35 


10.86 


55.70 


16.64 


14.14 


17.07 


12.94 


60.79 


158.62 


5.11 


12.06 


17.17 


36 


11.73 


9.15 


10.93 


5.92 


37.73 


14.59 


13.22 


16.50 


11.44 


55.75 


17.60 


14.90 


19.23 


14.77 


66.50 


159.9S 


5.29 


11.10 


16.39 


37 


12.00 


8.86 


10.00 


7.50 


38.36 


15.27 


13.18 


15.17 


11.17 


54.79 


16.83 


14.05 


16.90 


12.08 


59.86 


153.01 


4.40 


8.21 


12.61 


38 


12.87 


9.83 


11.40 


7.36 


41.16 


14.93 


12.83 


15.25 


11.90 


54.91 


16.70 


14.64 


18.29 


13.05 


62.68 


159.05 


4.89 


10.81 


15.70 


39 


11.43 


9.17 


11.56 


7.38 


39.54 


15.50 


13.29 


16.20 


12.22 


57.21 


17.70 


15.00 


18.55 


14.35 


65.60 


162.35 


6.79 


10.30 


17.09 


40 


11.75 


9.65 


11.21 


7.00 


39.61 


15.03 


12.33 


16.00 


11.25 


54.61 


16.39 


14.50 


18.35 


12.70 


61.94 


156.16 


4.94 


8.31 


13.25 


41 


12.61 


9.85 


11.34 


9.24 


43.04 


15.44 


12.56 


15.10 


12.29 


55.39 


16.67 


14.55 


18.21 


13.28 


62.71 


161.14 


6.32 


9.57 


15.89 


42 


12.62 


9.50 


10.96 


7.S3 


40.91 


15.00 


12.94 


15.88 


12.50 


56.32 


17.38 


15.16 


19.81 


15.40 


67.75 


164.98 


5.34 


9.85 


15.19 


43 


11.89 


9.57 


11.44 


S.13 


41.03 


15.00 


12.70 


15.67 


10.93 


54.30 


15.50 


13.00 


17.83 


13.00 


59.33 


154.66 


5.34 


8.50 


13.84 


45 


12.65 


9.66 


11.72 


7.33 


41.36 


14.81 


11.88 


15.31 


11.20 


53.20 


16.70 


14.40 


17.75 


13.30 


62.15 


156.71 


5.64 


9.65 


15.29 


47 


13.28 


9.86 


11.67 


9.39 


44.20 


15.80 


13.69 


17.21 


11.83 


58.53 


17.31 


13.6S 


18.11 


12.92 


62.02 


164.75 


4.93 


8.54 


13.47 


48 


11.41 


8.88 


10.39 


6.27 


36.95 


13.40 


10.00 


11.89 


7.70 


42.99 


15.25 


13.25 


17.30 


11.80 


57.60 


137.54 


2.96 


6.95 


9.91 


14 


11.38 
10.00 
12.14 
10.94 
11.42 
13.40 
10.00 
11.50 
12.50 
12.14 


9.36 
8.77 
9.04 
9.13 
9.06 
10.19 
9.27 
9.44 
9.4S 
9.39 


11.83 
10.14 
11.39 
10.60 
10.80 
11.33 
10.S3 
9.43 
11.45 
11.29 


6.64 

5.80 
6.10 
5.83 
8.10 
S.00 
7.38 
6.38 
8.50 
son 


39.21 
34.71 
38.67 
36.50 
39.38 
42.92 
37.4S 
36.75 
41.93 
40.8S 


16.25 
15.00 
13.88 
14.17 


13.35 
12.75 
10.94 
11.13 


16.5S 
16.50 
13.92 
14.60 


12.20 
11.50 
U.OO 
10.64 


58.3S 
55.75 
49.74 
50.54 












97:59 
90.46 
88.41 
87.04 
39.38 
42.92 
37.48 
36.75 
41.93 
40.88 


4.84 
6.18 
5.76 
4.78 




4.84 


21 












6.18 


26 








5.76 


34 








4.78 


7 
















29 
























44 






















46 




















50. 














51 














A V.ldi 


ion. 


S— ! 


Subtract 


on. 


M— I 


lultiplic 


ation. 


D- 


-Din'sioi 


. 





















14 PUBLIC SCHOOLS OF NEWARK, N. J. 

numbe- 36 is 40th in 4A, 20th in 6A and 2nd in 8A. These schools show 
a commendable improvement, but they also show that the work of the 
primary grades is not up to the average. On the other hand, the 
comparative standing of the grades of other schools, presents a reverse 
condition. This occurs less frequently. For example, school number 9 
begins as 1st in 4A, is 15th in 6A, and 16th in 8A. School number 47 
is 2nd in 4A, 3rd in 6A, and 23rd in 8A. While there is a decrease in 
efficiency from the 4A to the 8A grade in these schools, the work of the 
primary grades was exceptionally well done. The results in the reason- 
ing test follow in a large measure, the results in the four fundamentals. 
A school which loses position in the four fundamentals as it moves 
toward the higher grades is not able to show results in reasoning. 

Some schools vary decidedly with reference to. processes. Schools 
number 22 and 36 are weak in addition; numbers 1, 15, and 31, in sub- 
traction; numbers 5, 8, 9, 13, 28, 35, and 41, in multiplication; and num- 
bers 16, 27, 33, and 47, in division. Many of these schools show great 
variance also by grades. 

The analytic processes are naturally the reverse of the synthetic 
processes and follow them. There are, however, several schools where 
the results in the analytic processes are much better than those in the 
synthetic, as in schools numbers 9, 13, 22, and 28. This condition is 
illustrated by giving the rank of these schools in each of the four funda- 
mentals and grouping them under synthetic and analytic processes, as 
follows : 





Synthetic 


Analytic 


School 


Addition Multiplication 


Subtraction Division 


9 


7 13 


3 1 


13 


27 30 


5 13 


22 


21 7 


11 4 


28 


18 25 


2 19 



Totals 73 75 21 



More attention to the synthetic processes might have materially 
raised the general standing of these schools. 

On the other hand, in schools numbers 16, 19, 25, 31, 33, and 47, 
the results in the synthetic processes are far better than in the analytic. 
The rank of these schools in the two processes is as follows : 







Sj 


nthetic 


Analytic 


School 


Addition 


Multiplication 


Subtraction Division 


16 


5 




22 


18 29 


19 


14 




17 


25 23 


25 


10 




3 


17 18 


31 


26 




9 


37 28 


33 


1 




1 


6 15 


47 


2 




4 


7 12 



Totals 58 56 110 125 

These schools could readily profit by giving more attention to the 
analysis of the synthesis, although from two investigations subtractive 
subtraction was shown to be better than additive subtraction. 



ARITHMETIC SURVEY 15 

• It will be seen from Table I that school number 18 stands first, 
without a close competitor. A careful analysis of this school's work 
places it as 14th in addition, 3rd in subtraction, 14th in multiplication, 
and 8th in division in the 4A grade ; 4th in addition, 8th in subtraction, 
3rd in multiplication, and 1st in division, in the 6A grade; 3rd in addi- 
tion, 3d in subtraction, 2nd in multiplication, and 5th in division, in the 
8A grade; making it stand 7th in 4A, 1st in 6A, and 3rd in the 8A 
grades; also 5th in addition, 2nd in subtraction, 1st in multiplication, 
and 3rd in division, in all three grades. This gives it first place in the 
four fundamental operations in the whole city. In the reasoning test 
the. school stands 2nd in 6A and 1st in 8A, making it first in reasoning. 
There are a number of schools that stand very high in the list. 
Attention is called to numbers 18, 23, 12, 33, 39, 22, 11, 28, 10, and 41, 
as the ten that stand highest in the combination of results in funda- 
mentals and reasoning. In only one of these schools does the compar- 
ative standing in reasoning fall below that in the fundamentals. On 
the other hand, numbers 48, 2, 3, 37, 6, 20, 30, 24, 5, and 8; represent the 
ten lowest. In five of these ten .schools the comparative standing in 
reasoning is lower than in the fundamentals. The correlation between 
the comparative standing in fundamentals and the comparative standing 
in reasoning is very close. While there is naturally correlation between 
them in the work of individuals due to mental ability, one fails to see 
how this natural correlation can exist when schools are compared, where 
all kinds of minds are supposed to be found. This same condition 
exists when we take any group of schools presenting the highest stand- 
ard and compare it with a similar group with a correspondingly low 
standard. The conclusion then can fairly be drawn that pupils do not 
learn to reason well until they have mastered the machinery by which 
they work out their steps in the reasoning process. 

For the purpose of comparing types of schools, the schools have 
been grouped as follows : Those with pupils of Italian parentage, those 
of Hebrew parentage, those of prosperous Americans, those of less 
prosperous Americans, alternating schools, and all year schools, with the 
results as indicated in Table II. This table contains the sum of the 
scores in the four fundamentals in the 4A and 8A grades, and the per 
cent, of improvement from 4A to 8A ; the score in reasoning in the 6A 
and 8A grades and the improvement found between these grades. To 
this is added, for purposes of comparison, the average age of graduation 
in the various groups for the school year 1918-19 and the same records 
for the whole city. 

TABLE II 
Comparison of Groups of Schools 

Average age 
Accuracy Reasoning of graduates 

No. of Sum of Medians Per Cent. Averages Per Cent. 1918-19 

Schools 4A 8A Imp. 6A 8A Imp. Yrs. Mos. 

Italian 6 38.57 59.13 S3 3.91 7.80 99 13 11 

Hebrew 7 41.42 63.35 53 5.01 8.91 78 14 3 

Prosperous American 9 41.03 62.21 52 5.70 10.33 81 14 4 

Less Prosoerous American 6 39.49 62.04 57 5.28 9.42 78 14 4 

Alternating 8 38.69 60.44 56 4.21 8.57 104 14 1 

All Year - 5 38.80 58.32 50 3.82 7.61 99 *13 10 

Whole City 40.18 62.31 55 5.27 9.60 82 14 2 

* On account of the epidemic the first class graduated from the all year schools during the 
school year 1918-19 was in January, 1919, making the average nearly one month older than the two» 
following classes. 



16 PUBLIC SCHOOLS OF NEWARK, N. J. 

It will be seen that the Hebrew schools stand in the 4A in the 
highest position and finish in the 8 A in the highest position, making 53% 
increase in fundamentals, while in reasoning they stand third highest in 
the 6A and finish third highest in the 8A, making a moderate per cent, 
of increase. The average age of graduates of this group is one month 
higher than for the whole city. They appear to be especially strong in 
fundamentals but this strength is not maintained in reasoning, where 
they are surpassed by both groups of Americans. They appear to give 
closer attention to formal work (as shown also by the writing survey) 
than the Americans, but cannot reason as well. This group contains 
one alternating and one all year school, and is to that extent affected 
by them. 

The Italian schools start lowest and finish next to the lowest with 
a small per cent, of increase in the fundamentals. The results in rea- 
soning are also very poor, although the per cent, of increase in reasoning 
is large partly due to the fact that their score in the 6A is very low. The 
8A reasoning is next to the lowest. The average age of the graduates of 
this group is three months below the average for the city and the lowest 
of all except all year schools. This result may be affected by the fact 
that this group contains three alternating and four all year schools. 

The contrast between the schools of the prosperous American and 
those of the less prosperous American children is somewhat surprising. 
The schools of the less prosperous American children begin below those 
of the prosperous American children and finish below, but make a 
greater degree of progress in fundamentals, while in reasoning the same 
thing is repeated except the degree of progress is not so great in the 
schools having the less prosperous American children as in the schools 
having the prosperous American children. We cannot escape the con- 
clusion that there are a sufficient number of less prosperous Americans 
residing in these districts, who are less prosperous because of inferior 
mentality, to influence the results of these tests. Pupils of this type are 
frequently difficult to control. Many of them come from homes, where 
from various causes the conditions are such as not to contribute to a 
high standard of efficiency and the whole school is influenced by them. 
Because of the conditions enumerated above, many of the best teachers 
seek transfers to other schools. The group of schools of the prosperous 
American children finishes slightly below the city average in 8A in 
fundamentals and the highest of any group in the 8A reasoning. This 
group contains one alternating school but no all year school, while the 
average age of graduates is two months higher than the city average. 
The group of schools of the less prosperous American children contains 
neither alternating school nor all year school. 

The group of alternating schools begins lowest in the fourth grade 
and finishes third from the lowest in fundamentals, while in reasoning it 
begins third from the lowest and finishes third from the lowest. In this 
group of eight we have three Italian schools, two of the five all year 
schools and one school of the prosperous Americans. The average age of 
graduates is two months lower than the city average influenced largely 
by the Italian and all year schools. 



ARITHMETIC SURVEY 1/ 

All year schools begin low in the 4A in fundamentals, and finish 
lowest in the 8A, making only 47% of progress, while in reasoning they 
begin lowest, finish lowest and make 110% of progress, due, largely, to 
the low start. This group of schools contains three Italian and one 
Hebrew school while the average age of graduates is four months lower 
than the city average. 

In general, the high record in 8A in fundamentals is perhaps due 
largely to a desire on the part of the pupil to do careful work and on the 
part of the teacher to require efficient results. The schools composed 
largely of Hebrew children and the schools composed largely of Ameri- 
can children show the best results. Mentality, age or foreign parentage 
affects the work in the four fundamentals much less than in reasoning. 

Reasoning, since it involves the interpretation of printed matter, 
must be influenced by any lack of ability to> understand English or by 
mentality. The American pupils, therefore, should excel, but there 
appears to be no good reason other than the intelligence of the pupils 
for the standing in both the fundamentals and reasoning of the pupils of 
the all year and Italian schools compared with those of the Hebrew 
schools. The results in the all year and alternating schools may be 
influenced by the predominance of pupils of Italian parentage and 
because the thorough organization of these schools is not yet completed. 



DISTRIBUTION OF PUPILS' SCORES 

Chart IV shows the distribution by grades of pupils' scores in addi- 
tion and multiplication compared with the distribution by grades of 
pupils' scores in the reasoning test. The results of the work in the four 
fundamentals are not as diversified as are the results in the reasoning 
test. This is natural and leads to the conclusion that mental ability is 
not as important in securing results in the four fundamentals as in 
reasoning, and that we have in each grade pupils who may do quite 
uniform work in the fundamentals but who cannot do uniform work in 
reasoning. 

The improvement from grade to grade is not uniform in any case. 
In addition the greatest improvement occurs in 5A, in multiplication 5A, 
and in reasoning, perhaps 8A. The distance between these lines may 
be due to grading, or to the standard set by the course of study. 

The chart also shows that a great degree of proficiency in funda- 
mentals may be attained early in the grades and that the subsequent 
improvement is slight for each grade. The reasoning ability or interpre- 
tation of written problems appears to develop later and as the pupils 
become older and reach the higher grades the improvement by grades is 
very marked, as shown by the lines indicating the grade averages. 

It is interesting to note that many pupils of the lower grades can 
do better work than others in the upper grades. Two-tenths of one per 
cent, of the pupils in the 4A do as good work in addition as the average 
8A pupil and 17% in the 6A do as good work as the average 8A pupil ; 



Chart IZ 
Distribution ot ■Scores Attained in Addition, /lu/tip/icat/on and Reasoning 




18 



PUBLIC SCHOOLS OF NEWARK, N. J. 



<.S ^ 



N 



1 



I 
"5 



■5 fcX'> 




§^ 



N. 



1 



4°- 



~* 



ARITHMETIC SURVEY 19 

while 3.6% of the 8A pupils do no better work than the average 4A 
pupil and 36.2% do no better than the average 6A pupil. In reasoning 
5.5% of the 6A pupils could do as well as the 8A pupils while 9% of 
the 8A pupils can reason no better than the average 6A pupil. 



ACCURACY 

The accuracy of the work in the four fundamentals by grades is 
shown in the following table : 



TABLE III 

Per Cent, of Accuracy in the Four Fundamentals 

Per Cent. Correct on Total Attempts 



Grade 


Addition 


Subtraction 


Multiplication 


Division 


Average 


4A 


68.9. 


61.3 


60.3 


56.1 


62.2 


5B 


71.1 


72.2 


63.2 


64. 


67.6 


SA 


75. 5 


76.4 


74.8 


69.8 


74.1 


6B 


77.7 


79. 


74.2 


71.1 


75.5 


6A 


79.9 


79.6 


78. 


72.9 


77.6 


7B 


80.2 


82.6 


78.6 


74.4 


78.9 


7A 


82.6 


85.7 


84.3 


78.6 


82.3 


8B 


84. 


87.4 


84. 


82.8 


84.5 


8A 


85.7 


90.6 


88.1 


84.9 


87.3 


Total City 


76.8 


76.1 


73.8 


70.2 


74.2 



(See Charts II and V for graphs showing the average per cent, of accuracy by grades.) 

Chart V shows the per cent, of accuracy graphically. It will be 
noticed that accuracy in addition improves the least of any of the four 
fundamentals. This is probably due to the fact that addition examples of 
medium length do not often occur in our books in arithmetic, nor is there 
a strong motive created to secure accurate results in this one of the four 
fundamentals, while the other processes more frequently occur in sub- 
sequent work. 

The per cent, of accuracy in the reasoning test based on the number 
of problems reasoned correctly is 92.9% in the 6A, and 94.3% in the 
8A ; while the per cent, of accuracy based on the total number of 
examples attempted is 57.9% in 6A, and 74.8% in 8A. The manipulation 
of the figures involved in these problems is very simple indeed, so that 
if pupils understand what to do they can very easily do it. The test is 
based almost entirely on the interpretation of the written problem and 
not the working: of it. 



DISCUSSION OF INDIVIDUAL PROBLEMS 

A brief review of the errors made in the individual problems of these 
tests may serve to emphasize some of the weak points they have brought 
to light. 

The Woody Scale for each of the four fundamental operations is 
printed, followed in each case by a table showing the number of times 
each problem was worked incorrectly or not attempted. (See Tables 



20 



PUBLIC SCHOOLS OF NEWARK, N. J. 



IV, V, VI, and VII.) The number of times each problem was worked 
correctly can easily be found. The Stone Reasoning Test is also printed, 
followed by a discussion of the results obtained on each individual 
problem. 













Addition Scale 






(1) 


(2) 






(3) 


(5) 


(7) 


(ID 


2 


2 






17 


72 


3 + 1 = 


21 


3 


4 






2 


26 




33 


— 


3 






— 


— 




35 


(13) 




(14) 






(16) 


(19) 


(20) 


23 




25 + 42 = 






9 


$ .75 


$12.50 


25 










24 


1.25 


16.75 


16 










12 
15 
19 


.49 


15.75 
















(21) 




(22) 






(23) 


(24) 


(30) 


$8.00 




547 






Vz + y 3 = 


4.0125 


2J4 


5.75 




197 








1.5907 


6H 


2.33 




685 








4.10 


3 34 


4.16 




678 








8.673 





.94 




456 
















6.32 




393 

525 
240 






























152 












(33) 






(36) 




(38) 




.49 




2yr. 


5 


mo. 


25.091 + 


100.4 + 25 + 98.28 + 


19.3614 = 


.28 




3 yr. 


6 


mo. 








.63 




4 yr. 


9 


mo. 








.95 




5 yr. 


2 


mo. 








1.69 




6 yr. 


7 


mo. 








.22 
.33 






























.36 
















1.01 
















.56 
















.88 
















.75 
















.56 
















1.10 
















.18 
















.56 

















In examining the results of the examples set in the addition scale, it 
was found that the answer to the first example in addition was frequently 
given as 6, which might have resulted either by multiplying or by adding 
the number of the example with the other figures. In example 7, where 
the example was stated horizontally, the sign was very frequently dis- 
regarded. Thirty-seven per cent, of the pupils in the 4A, 14% of those 
in the 6A, and; 21% of all pupils failed to work it correctly. Failure to 
observe the signs when the example was stated horizontally was also 
shown in example 14, where 19.3% of all grades failed. Table IV shows 
that many pupils did. not attempt No. 14. The percentage of failures in 
numbers 7, 14, and 16 seem to indicate that these examples presented 
exceptional difficulties to our pupils. (The examples in the Series B 
tests are supposed to present equal steps of difficulty.) In example 16 
many pupils obtained the answer 70, having omitted the figure 9 at the 
top of the column. The answer to example 23 was frequently given 
as I. In the 8A, 51% failed in example 36, 37% in example 33, and 
25% in example 30. Example 38 proved to be very difficult for all 



ARITHMETIC SURVEY 



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PUBLIC SCHOOLS OF NEWARK, N. J. 







(27) 






5 


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grades. This example, of course, should not be attempted by pupils 
below the 5A grade. In many cases it was recopied in column form 
and it was very rare that the answer was correct where this was not 
done. Fifty-eight per cent, of the 8A pupils failed on this example. 

In the 4A there were 15 zero papers, one perfect paper, and one with 
18 of the 19 examples correct. In the 6A grade there were two zero 
papers, and 24 perfect papers ; in the 8A one with a score of 2, one with 
a score of 8, and 92 perfect papers, or 8% perfect. 



Subtraction Scale 

(1) (3) (6) (7) 

S 2 11 13 

_5 1 7 8 

(9) (13) (14) (17) 

7 « 16 50 393 

3J_ 9 25 178 

(19) (20) (24) (25) 

567482 234 — 1— 8% 27 

106493 5U 1254 

(31) (35) 

7.3—3.00081= 3Vs — 154 = 



One of the difficulties in the results of this scale was that many 
pupils added instead of subtracted, although they were told at the begin- 
ning that these were subtraction examples, and the word "subtraction" 
was printed at the top of the sheet. 

Examples 17 and 19 were wrong in 9% and 16% of the cases, 
respectively. In example 20 there were 29% of failures, many of them 
due to a complicated system of changing 1 to fourths and in the compli- 
cation the pupils became lost. There were many evidences of work of 
this kind, showing that the pupils had not had a sufficient amount of oral 
work. Many pupils passed over this example to do other more difficult 
ones. (See Table V.) 

In example 24, 9% of the 8A pupils and 45.5% of all pupils failed. 
In example 25, 18% of the 8A and 56.5% of all pupils failed. In example 
27, subtraction of compound denominate numbers, 29% of the 8A's and 
67.8% of all pupils failed. Fifty-five per cent, of all pupils failed in the 
31st example. In example 35, 24% of the 8A's and 68.9% of all pupils 
failed. Table V shows that although many avoided example 31 and 
went to example 35, the percentage of error in example 35 exceeded that 
in example 31. The principal cause of the error in example 35 was that 
when the answer was given as 2t. according to the way in which 
Mr. Woody formed his standards, it was counted an error. 

In the 4A there were 93 zero papers., and the scores of two of the best 
papers were 13 out of a total of' 15. The 6A contained two zero scores, 
and 159 perfect papers, or 9%. The 8A had four papers with a score of 
8 each, and 342 perfect papers, or 30% perfect. 



ARITHMETIC SURVEY 



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24 PUBLIC SCHOOLS OF NEWARK, N. J. 

Multiplication Scale 



(1) 

3X7 = 




(3) 
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The lowest per cent, of error on any one example was found in 
number 3 of the multiplication scale, which was one-half of one per cent. 
The per cent, increased until it reached example 11, where 10% were 
wrong. Example 18 was incorrectly worked by 25.6% of all pupils, 
while 9% of the 8A grade failed in this example. The principal error in 
example 18 was the placing of the partial product incorrectly, making 
the first figure of the partial product from the multiplication by 2 fall 
under the first figure of the multiplication by 3. Examples 20 and 27 
were incorrectly worked largely on account of the omission or incorrect 
placing of the decimal point, the percentage of failures in grades above 
the fifth being 11% in the former and 26% in the latter. In the 20th ex- 
ample many pupils multiplied by zero, placing three zeros as a partial 
product ; and others multiplied by zero getting a product of the value of 
multiplication by one, thereby obtaining an answer of 43.05 instead of 
14.35. Examples 24 and 26 were the cause of 44% of failures each. In the 
26th example many multiplied by 9 only. This occurred in one school 
sixteen times in the 6A grade. Example 26 was avoided by more than 
twice as many pupils as example 24. In the eighth grade only were the 
non-attempts of the 24th less than the 26th, while the per cent, of errors 
was greater in the 26th than the 24th in all grades above the 5B. In 
example 29, 56% failed, which shows quite conclusively that pupils have 
not done a sufficient amount of this work orally. The whole line of 
examples beginning with the 24th and ending with the 29th was skipped 
in many cases. Examples 33 and 37 had a per cent, of error of 52% and 
71%, respectively, while the 8A grade failed in 19% of the cases in 
example 33 and 44% in example 37. The errors were made partly in 
reducing the mixed numbers, and partly by the inversion of one or more 
of the fractions. The principal error in the 35th example was caused by 
bringing down the }i instead of multiplying it by 25. The 38th example 
was incorrectly worked by 76% of all pupils, and 47% of the 8A grade. 
The same error of bringing down the fraction instead of multiplying 
it, and the misplacement of the decimal point were the chief causes for 
low records in this example. Much time was lost by working on scraps 
of paper and on the backs of the sheets. 



ARITHMETIC SURVEY 



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26 PUBLIC SCHOOLS OF NEWARK, N. J. 

In the 4A there were six zero scores, and the 23 highest papers had 
a score of 14 out of a total of 20. In the 6A there were two zero scores, 
and 13 perfect papers. The 8A contained one paper with a score of 4, 
two with a score of 10, and 122 perfect papers, or 10% perfect. 



( i) 

3 | 6 

( 11) 
2 ] 13 

(19) 
248^-7 = 

(30) 
«-5-5 = 

The work in this scale was the most inaccurate of any of the opera- 
tions. Beginning with example 2 with 1.5% of errors, the increase is 
quite uniform until we reach example 14, when the difficulty becomes 
quite apparent. The cause of failures in example 7 was generally due 
to inability of pupils to work examples expressed horizontally. Some 
resorted to a very complicated arrangement of figures, as 

2 

l 

More oral work with and without text would remedy this. Example 8 
presented considerable difficulty, the answers appearing in the following 
forms : 

Jl JL _i! ol= * 

9)0 9)0 9)0 9)0 

It is doubtful if this example presents a practical situation. Example 
14 caused 16% of errors; and examples 15 and 27 were the cause of 21% 
and 50% of errors, respectively. There were also an exceptionally large 
number of pupils who failed to attempt these two examples. These 
examples presented such great difficulties to the pupils that it was quite 
apparent they were not sufficiently familiar with them. One school 
with two 4A classes did not have a paper with the 15th example correct. 
Another school with 42 4A pupils had only seven correct; in another 
school only four 4A pupils attempted examples 15 and 27, and of these 
four attempts two had number 15 correct, one multiplied in number 15 
and the other divided number 27 by 8 twice. In another school no 
pupil in the 6A class obtained a correct answer for number 27. 

Example 17 was also curiously treated, the results being 7-1, 7sV, 
7(1), 7\ 7&>, 7-t-, 7" 5 T°"- I n many cases the pupils possessed the 
knowledge but did not express it as required by the Woody test. The 
common error in the 23rd example was failure to place a zero as the 



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28 PUBLIC SCHOOLS OF NEWARK, N. J. 

second figure of the quotient. The seventh and eighth grade pupils 
failed more frequently in this example than in the 27th. The errors in 
the 28th example seemed to be largely by schools. In a 6A class of one 
school, 24 out of 44 pupils worked it incorrectly. 

By multiplying in example 30 many gave 3fy as the answer. In 
some cases the errors were caused by their not knowing how to handle 
the 5, and in other cases by the inversion of the %. This example gave 
72% of failures, 63% in grades above the 5B and 46% in the 8A. In the 
36 example 80% of the 8A failed. Division of denominate numbers, 
however, is not to be found in our course of study, having been purposely 
omitted. 

The 4A contains eight zero papers, with five of the highest papers 
having a score of 12 out of a possible 15. The 6A contains three having 
a score of 2, and five perfect papers. The 8A has two with a score of 6, 
and 98 perfect papers, or 9% perfect. 

Reasoning Test 

Solve as many of the following problems as you have time for; work them in order as 
numbered: 

1. If you buy 2 tablets at 7 cents each and a book for 65 cents, how much change should you 
receive from a two-dollar bill ? 

2. John sold 4 Saturday Evening Posts at 5 cents each. He kept y 2 the money and with the 
other Yz he bought Sunday papers at 2 cents each. How many did he buy? 

3. If James had 4 times as much money as George, he would have $16. How much money 
has George? 

4. How many pencils can you buy for SO cents at the rate of 2 for 5 cents? 

5. The uniforms for a baseball nine cost $2.50 each. The shoes cost $2 a pair. What was 
the total cost of uniforms and shoes for the nine ? 

6. In the schools of a certain city there are 2,200 pupils; y 2 are in the primary grades, % in 
the grammar grades, % in the High School and the rest in the night school. How many pupils 
are there in the night school? 

7. If 3y 2 tons of coal cost $21, what will S l / 2 tons cost? 

8. A newsdealer bought some magazines for $1. He sold them for $1.20, gaining 5 cents on 
each magazine. How many magazines were there? 

9. A girl spent Y% of her money for car fare, and three times as much for clothes. Half of 
what she had left was 80 cents. How much money did she have at first? 

10. Two girls receive $2.10 for making button-holes. One makes 42, the other 28. How shall 
they divide the money? 

11. Mr. Brown paid one-third of the cost of a building; Mr. Johnson paid Yi the cost. Mr. 
Johnson received $500 more annual rent than Mr. Brown. How much did each receive? 

12. A freight train left Albany for New York at 6 o'clock. An express left on the same track 
at 8 o'clock. It went at the rate of 40 miles an hour. At what time of day will it overtake the 
freight train if the freight train stops after it has gone 56 miles? 

The Stone Reasoning Test presents an entirely different phase from 
that of the four fundamentals. The formal arithmetic required is very 
simple, most of the difficulty being due to the proper interpretation of 
the written problem. This test is constructed so that each problem offers 
increasing difficulties, and very few pupils are expected to finish in the 
time allowed — fifteen minutes. Every step in the problems was marked 
correct if reasoned correctly, whether the correct results were obtained 
or not. Most of the failures appeared to be due, first, to inability to 
reason, and second, to inability to interpret the printed statement, and to 
fully recognize the value of each word. It is quite evident from these 
test papers that teachers should train pupils to interpret oral and written 
statements more carefully. This can be done very easily in almost all 
subjects, or if necessary by specially prepared written or printed direc- 
tions, requiring on the part of the pupil silent interpretation followed by 



ARITHMETIC SURVEY 29 

action. It is very frequently the habit of teachers not to allow pupils to 
make mistakes, whereas it would be much better if the pupil actually 
made the mistake under the supervision of the teacher and then was 
brought face to face with the results of his own incorrect interpretation. 

In the Newark tests many pupils copied the questions before 
answering them ; others had an elaborate system of working out the 
formal arithmetic of the problem, and still others analyzed on scratch 
paper and then copied the work on the answer paper. Many pupils 
went to the other extreme and merely copied the answer on the final 
paper, doing the work either without figures or on scratch paper. In 
this way many failed to receive any credit for a partially correct example, 
as no work was shown on which any credit could be based. 

A very noticeable improvement in reasoning and in neatness was 
shown between the 6A and 8A grades. There were classes in which all 
pupils showed very commendable work as to reasoning and appearance. 
As a general rule, the papers that were neatly arranged were much more 
accurate than those that were carelessly arranged. This was also true 
in the work in fundamentals. The papers from some schools seemed to 
indicate that the dollar sign was considered a very unimportant factor. 
In fact, it seemed in some cases as though the children were not required 
to use it at all. In many cases the children could not spell the name of 
their school, and it was very common to find the name of the teacher 
misspelled. One teacher's name was spelled in eighteen different ways. 

In example 1 the word "each" was the cause of most of the failures, 
resulting in the pupils using 7^ as the cost of two tablets. In the third 
step of the second example many multiplied instead of divided. The 
fourth example was very poorly done; in many cases 50$ was divided 
by 2 and again by 5^. The fifth example presented very little difficulty ; 
when errors occurred they were generally due to placing the $18 in the 
cents column when adding it to $22.50. Many pupils worked the sixth 
example as though it read "J4 of the remainder and z /g of those then 
remaining." There were many errors, however, in getting % of 2,200. 
Very few errors occurred when the pupils added the fractions and sub- 
tracted from %. In the seventh example, many who failed expressed 
the first step 3^-^-21, and many divided in the second step instead of 
multiplied. In the eighth example, many expressed the first step in 
this form $1 — $1.20 = $.20 gain. 

The complications increased toward the end of the set and were 
mastered by only the best minds. In the ninth example the stumbling 
block was "half of what she had left." The reasoning here began to be 
wide of the mark. The errors in the tenth were generally due to the 
decimal point. Most pupils failing on this example had $3.00 or $.30 
for making one buttonhole. Some of the pupils were able to solve social 
questions better than mathematical questions, and suggested that each 
be given one-half of the money. One pupil suggested that each girl be 
given half of the buttonholes to work. The eleventh example was 
worked incorrectly in most cases — many completed the first step and 
stopped. In the twelfth, many pupils reasoned out \ 2 /<-> hours later, 
which was partially correct and for which full credit was given. 



30 



PUBLIC SCHOOLS OF NEWARK, N. J. 



Thirty-two of the 2,352 6A pupils received a score of zero on this 
test and 29 a score of less than one point out of a possible 17.2 points. 
Ninety-one of the 1,511 8A pupils received a score of less than 5 points. 

This discussion of individual problems seems to present a dis- 
couraging picture, but it is an array of errors only for the purpose of 
profiting by the knowledge of those errors. There were many excellent 
papers, many classes did highly commendable work, and many schools 
stood high. This must not be lost sight of. 



CONCLUSIONS 

From the foregoing the most important conclusions are : 

That of the four fundamentals the results in addition are poorest 
when compared with the Woody standard, caused probably by the lack 
of organization of this subject in the lower grades where the combina- 
tions are supposed to be learned, and by lack of subsequent rational and 
systematic drill in the higher grades. 

That the results in multiplication are the best of the four funda- 
mentals. 

That in some schools-the results in the analytic processes are much 
better than those in the synthetic processes. 

That the synthetic and analytic processes seem to be mutually 
beneficial. 

That the results in arithmetic in the primary grades are well up to 
the standard but that the improvement is not maintained in the middle 
grades. 

That the weak grades seem to be 6B, 6 A, and 7B, both as to funda- 
mentals and as to reasoning. The higher grades reclaim some of the 
lost ground. The cause of the loss should be a matter of study and 
investigation. 

That the results in reasoning as a whole are good and except for 
the loss in the 6A and 7B grades the work of the Newark schools 
deserves commendation. 

That schools which stand high in fundamentals also stand high in 
reasoning and vice versa. The reasoning seems to be very much 
impaired when the machinery with which the problems are worked is 
weak. 

That the "A" classes are stronger than the "B" classes, and that 
the 5A grade is the strongest of all tested. 

That accuracy necessarily accompanies any satisfactory amount of 
good work accomplished. 

That the improvement from grade to grade is not uniform. 

That there are pupils in the lower grades who can greatly surpass 
some in the upper grades in work which both have been taught. 

That the age of the pupils affects considerably the results of their 
work. 

That the work of the Italian, alternating, and all year schools is 
poor but that the age is lower than the average. 

That a good teacher affects very materially the results. 



